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Lesson 10 - QR Summary of Sections 1 thru 4

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Objectives Review Hypothesis Testing –Basics –Testing On Means, μ, with σ known Z 0 Means, μ, with σ unknown t 0 Population Proportion, p, Z 0

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Determining H o and H a H o – is always the status quo; what the situation is currently the claim made by the manufacturer H a – is always the alternative that you are testing; the new idea the thing that proves the claim false

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Reality H 0 is TrueH 1 is True Conclusion Do Not Reject H 0 Correct Conclusion Type II Error Reject H 0 Type I Error Correct Conclusion H 0 : the defendant is innocent H 1 : the defendant is guilty Type I Error (α): convict an innocent person Type II Error (β): let a guilty person go free Note: a defendant is never declared innocent; just not guilty decrease α increase β increase α decrease β Four Outcomes from Hypothesis Testing

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Three Ways – H o versus H a 1.Equal versus less than (left-tailed test) H 0 : the parameter = some value (or more) H 1 : the parameter < some value 2. Equal hypothesis versus not equal hypothesis (two-tailed test) H 0 : the parameter = some value H 1 : the parameter ≠ some value 3. Equal versus greater than (right-tailed test) H 0 : the parameter = some value (or less) H 1 : the parameter > some value b a b a Critical Regions

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Hypothesis Testing The process of hypothesis testing is very similar across the testing of different parameters The major steps in hypothesis testing are –Formulate the appropriate null and alternative hypotheses –Calculate the test statistic –Determine the appropriate critical value(s) –Reach the reject / do not reject conclusions –Always put pieces into context of the problem

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Similarities in hypothesis test processes ParameterMean ( σ known) Mean ( σ unknown) Proportion H0:H0:μ = μ 0 p = p 0 (2-tailed) H 1 :μ ≠ μ0μ ≠ μ0 μ ≠ μ 0 p ≠ p 0 (L-tailed) H 1 :μ < μ 0 p < p 0 (R-tailed) H 1 :μ > μ 0 p > p 0 Test statisticDifference Critical valueNormalStudent tNormal

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Using Your Calculator Press STAT –Tab over to TESTS –Tests of μ (population mean) For σ known -- Select Z-Test and ENTER For σ unknown – Select T-Test and ENTER –Tests of p (population proportion) Select 1-Prop Test and ENTER –Computed p-values can be directly compared to α –Computed test-statistic (z 0 or t 0 ) can be compared to appropriate critical value (gotten from tables)

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Example from 10-2 Problem 28 on page 529: a) The normality plot in the text shows that the data distribution appears normal. The box-plot of the data does not show any outliers, Assumptions Met. b)H 0 : μ = 0 (Summer of 2000 average temperature does not differ from an average years temperature) H a : μ > 0 (Summer of 2000 is hotter than usual) Since we have raw data, we need to enter our data (12 values) into L1 in our calculator before we attempt the hypothesis test. After we enter our data, we then go to z-test (because σ is given to us in the problem σ known) Calculator Input: Calculator Output: (of interest) μ 0 : 0 σ : 1.8 z = 1.9245 List : L1 p = 0.02715 Freq : 1 > μ 0

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Example from 10-2 cont Problem 28 on page 529: b) Calculator output of interest: z = 1.9245 p = 0.02715 P-Value Method: Compare p-value with α if p-value z c then we Reject H 0. We look into the table for.95 and come up with a z c = 1.645 [or we use our calculator to find it invNorm(0.95) = 1.64485]. Since z 0 > z c we reject H 0. Confidence Interval Method: Since this is a right tailed test and not a two- tailed test, we would not use this method.

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Example from 10-3 Problem 19 on page 539: a)H 0 : μ = 40.7 (Average age of death-row inmates) H a : μ ≠ 40.7 (Average age of death-row inmates has changed) Check requirements: SRS normality (have to assume because we only have summary data) outliers (same as above) Since we have only summary data, we can go directly to the hypothesis test. We use t-test (because σ is not given to us in the problem σ unknown). We have a two-tailed test (different) from the problem. Calculator Input: Calculator Output: (of interest) μ 0 : 40.7 x-bar : 38.9 s x : 9.6 t = -1.0607 n : 32 p = 0.29704 μ : ≠ μ 0

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Example from 10-3 cont Problem 28 on page 529: a)Calculator output of interest: t = -1.0607 p = 0.29704 P-Value Method: Compare p-value with α if p-value 0.05 therefore we fail to reject H 0 and conclude that we don’t have sufficient evidence to say the average of death-row inmates has changed. Classical Method: Compare t statistic with t critical (from table), if t 0 > t c then we Reject H 0. We look into the table for t 0.025,31 and come up with a t c = 2.04. We add a negative sign to t c. Since t 0 < t c we fail to reject H 0. b)Confidence Interval Method: We can use the formula below to figure out the lower (LB) and upper bounds (UB) to compare with x-bar. u 0 +/- t c ∙ s x / n Or we can use our calculator, Stat Tests t-interval Enter the same data we had for the t-test before Calculator Output: (35.439, 42.361) Since μ 0 lies in the interval, we fail to reject H 0.

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Example from 10-4 Problem 12 on page 551: a)H 0 : p 0 = 0.33 (American adults who believe in haunted houses) H a : p > 0.33 (Percentage of American adults who believe in haunted houses has increased) Check requirements: SRS 1004 American adults 10 1004(0.33)(0.67) = 221.98 >> 10 Since we have a proportion problem, we need to use the 1-Prop ZTest in our calculator. We are doing a right tailed test (% has increased). Calculator Input: Calculator Output: (of interest) p 0 : 0.33 x : 370 z = 2.596 n : 1002 p = 0.0047 prop : > p 0

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Example from 10-4 cont Problem 12 on page 551: a)Calculator output of interest: z = 2.596 p = 0.0047 P-Value Method: Compare p-value with α if p-value z c then we Reject H 0. We look into the table for.95 and come up with a z c = 1.645 [or we use our calculator to find it invNorm(0.95) = 1.64485]. Since z 0 > z c we reject H 0. Confidence Interval Method: Since this not a two tailed test, we normally would not use this method.

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Summary and Homework Summary –We can test whether sample data supports a hypothesis claim about a population mean, proportion, or standard deviation –We can use any one of three methods The classical method The P-Value method The Confidence Interval method (two-tailed tests) –The commonality between the three methods is that they calculate a criterion for rejecting or not rejecting the test statistic Homework –pg 511-513; 1, 2, 3, 7, 8, 12, 13, 14, 15, 17, 20, 37

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